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Generalized Jacobi Method for Computing Eigenvalues of Dual Quaternion Hermitian Matrices

Yongjun Chen, Liping Zhang

TL;DR

The paper tackles the problem of computing all eigenvalues and eigenvectors of dual quaternion Hermitian matrices by extending the Jacobi eigenvalue method. It develops three Jacobi-type algorithms, including a novel three-step scheme (3SJacobi), together with convergence analyses and an acceleration strategy. The results show that these methods yield $O(\,\epsilon)$-approximate diagonalizations in finite iterations and can handle cases where eigenvalues share the same standard parts but differ in dual parts, outperforming the power method and Rayleigh quotient iteration in numerical experiments. The work advances stable spectral theory for dual quaternion matrices with practical implications for robotics and multi-agent systems, offering scalable and robust eigenvalue computations. Throughout, all mathematical notation is presented with precise $...$ formatting.

Abstract

Dual quaternion matrices have various applications in robotic research and its spectral theory has been extensively studied in recent years. In this paper, we extend Jacobi method to compute all eigenpairs of dual quaternion Hermitian matrices and establish its convergence. The improved version with elimination strategy is proposed to reduce the computational time. Especially, we present a novel three-step Jacobi method to compute such eigenvalues which have identical standard parts but different dual parts. We prove that the proposed three-step Jacobi method terminates after at most finite iterations and can provide $ε$-approximation of eigenvalue. To the best of our knowledge, both the power method and the Rayleigh quotient iteration method can not handle such eigenvalue problem in this scenario. Numerical experiments illustrate the proposed Jacobi-type algorithms are effective and stable, and also outperform the power method and the Rayleigh quotient iteration method.

Generalized Jacobi Method for Computing Eigenvalues of Dual Quaternion Hermitian Matrices

TL;DR

The paper tackles the problem of computing all eigenvalues and eigenvectors of dual quaternion Hermitian matrices by extending the Jacobi eigenvalue method. It develops three Jacobi-type algorithms, including a novel three-step scheme (3SJacobi), together with convergence analyses and an acceleration strategy. The results show that these methods yield -approximate diagonalizations in finite iterations and can handle cases where eigenvalues share the same standard parts but differ in dual parts, outperforming the power method and Rayleigh quotient iteration in numerical experiments. The work advances stable spectral theory for dual quaternion matrices with practical implications for robotics and multi-agent systems, offering scalable and robust eigenvalue computations. Throughout, all mathematical notation is presented with precise formatting.

Abstract

Dual quaternion matrices have various applications in robotic research and its spectral theory has been extensively studied in recent years. In this paper, we extend Jacobi method to compute all eigenpairs of dual quaternion Hermitian matrices and establish its convergence. The improved version with elimination strategy is proposed to reduce the computational time. Especially, we present a novel three-step Jacobi method to compute such eigenvalues which have identical standard parts but different dual parts. We prove that the proposed three-step Jacobi method terminates after at most finite iterations and can provide -approximation of eigenvalue. To the best of our knowledge, both the power method and the Rayleigh quotient iteration method can not handle such eigenvalue problem in this scenario. Numerical experiments illustrate the proposed Jacobi-type algorithms are effective and stable, and also outperform the power method and the Rayleigh quotient iteration method.
Paper Structure (13 sections, 14 theorems, 137 equations, 1 figure, 2 tables, 2 algorithms)

This paper contains 13 sections, 14 theorems, 137 equations, 1 figure, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Dual quaternions
  4. Dual quaternion matrices
  5. The Generalized Jacobi Eigenvalue Algorithm
  6. Jacobi method for dual quaternion Hermitian matrices
  7. Improved Jacobi method
  8. Numerical Experiments
  9. Performance of Algorithm \ref{['Jacobi_method_algorithm1']} and Algorithm \ref{['Jacobi_method_algorithm2']}
  10. Performance of Algorithm \ref{['Jacobi_method_algorithm_3']}
  11. On randomly generated dual quaternion Hermitian matrices
  12. On Laplacian matrices of randomly generated graphs
  13. Final Remarks

Key Result

Proposition 2.1

If $\hat{\mathbf Q}_1,\hat{\mathbf Q}_2\in \hat{\mathbb{H}}^{n}$, then we have where $\lambda(\hat{\mathbf Q}_1)=(\lambda_1(\hat{\mathbf Q}_1),\ldots,\lambda_n(\hat{\mathbf Q}_1))^\top$ and $\lambda(\hat{\mathbf Q}_2)=(\lambda_1(\hat{\mathbf Q}_2),\ldots,\lambda_n(\hat{\mathbf Q}_2))^\top$ with $\lambda_1(\hat{\mathbf Q}_1)\geq\lambda_2(\hat{\mathbf Q}_1)\geq\ldots\geq\lambda_

Figures (1)

  • Figure 1: Numerical performance of Algorithm \ref{['Jacobi_method_algorithm1']} and Algorithm \ref{['Jacobi_method_algorithm2']}.

Theorems & Definitions (38)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • Definition 2.10
  • ...and 28 more